View source for Weierstrass criterion (for uniform convergence)

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A theorem which gives sufficient conditions for the [[uniform convergence]] of a [[series]] or sequence of functions by comparing them with appropriate series and sequences of numbers; established by K. Weierstrass . If, for the series

$$ 
\sum _ { n= 1} ^  \infty  u _ {n} ( x)
$$

of real- or complex-valued functions defined on some set  $  E $
there exists a convergent series of non-negative numbers

$$ 
\sum _ { n= 1} ^  \infty   a _ {n}  $$

such that

$$ 
| u _ {n} ( x) |  \leq   a _ {n} ,\ \ 
n = 1, 2 \dots
$$

then the initial series converges uniformly and absolutely on the set  $  E $
(cf. [[Absolutely convergent series]]). For instance, the series

$$ 
\sum _ { n= 1} ^  \infty  

\frac{\sin  nx }{n  ^ {2} }

$$

converges uniformly and absolutely on the entire real axis, since

$$ 
\left | 
\frac{\sin  nx }{n  ^ {2} }
 \right |
 \leq   
\frac{1}{n  ^ {2} }

$$

and the series

$$ 
\sum _ { n= 1} ^  \infty   \frac{1}{n  ^ {2} }
$$
is convergent.

If, for a sequence of real- or complex-valued functions  $  {f _ {n} } $, 
$  n = 1, 2 \dots $
which converges to a function  $  f $
on a set  $  E $
there exists a sequence of numbers  $  \alpha _ {n} $, 
$  \alpha _ {n} > 0 $
and  $  \alpha _ {n} \rightarrow 0 $
as  $  n \rightarrow \infty $, 
such that  $  | f( x) - {f _ {n} } ( x) | \leq  \alpha _ {n} $, 
$  x \in E $, 
$  n = 1, 2 \dots $
then the sequence converges uniformly on  $  E $. 
For instance, the sequence

$$ 
f _ {n} ( x)  =  1 - \frac{(- 1)  ^ {n} }{x  ^ {2} + n }

$$

converges uniformly to the function  $  f( x) = 1 $
on the entire real axis, since

$$ 
| 1- f _ {n} ( x) |  <  
\frac{1}{n}

\  \textrm{ and } \  \lim\limits _ {n \rightarrow \infty } \ 

\frac{1}{n}
  =  0 .
$$

The Weierstrass criterion for uniform convergence may also be applied to functions with values in normed linear spaces.

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  K. Weierstrass,   "Abhandlungen aus der Funktionenlehre" , Springer  (1866)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  K. Weierstrass,   "Math. Werke" , '''1–7''' , G. Olms &amp; Johnson, reprint  (1927)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,   "Mathematical analysis" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Knopp,   "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR></table>